#27293959

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11620

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2966	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	#121×C₅₃	C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2966	D₅₃
hypocenter	C₅₃	#11620
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2966

318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,57,61)(55,58,60,59)(62,65,67,68)(63,70,71,77)(64,74,76,78)(66,79,81,82)(69,83,73,85)(72,86,87,90)(75,89,80,91)(84,92,88,93);(54,55)(56,58)(57,60)(59,61)(62,64)(63,69)(65,78)(66,80)(67,76)(68,74)(70,85)(71,73)(72,88)(75,81)(77,83)(79,89)(82,91)(84,87)(86,92)(90,93);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55,56,58,57,60,61,59)(62,63,66,72,68,70,82,86,67,71,81,87,65,77,79,90)(64,73,75,88,74,83,89,92,76,69,80,84,78,85,91,93)

Polycyclic presentation:

2^8,53|a^2=e,c^2=f,d^2=fh,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,i^a=i^52,c^b=ch,d^b=dfh,e^b=eh

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79435);